Russian conspiracy allegations will continue to dog the White House until President Trump gets a handle on the administration’s opponents inside the U.S. intelligence community who are driving the story, warned Republican strategists in Washington.

President Trump has railed against the leaks apparently emanating from the National Security Agency, CIA or FBI. But so far, he has failed to take adequate steps to root out the faction within the intelligence apparatus that is undermining his presidency, whether they are holdovers from the Obama administration or elements intent on thwarting Mr. Trump to preserve their own power.

The targeted leaks to the news media about contacts between the Trump campaign and Russian officials — including information from communications intercepted by intelligence agencies — resulted in the ouster of retired Lt. Gen. Michael Flynn as the president’s national security adviser and created a swirl of controversy last week around Attorney General Jeff Sessions.

The intelligence leaks have kept the story alive, fueled attacks by Democrats and fractured the Republican majority on Capitol Hill, although no evidence has emerged of actual collusion with Russians to affect the election.

To break the cycle, Mr. Trump must launch an internal investigation to get ahead of the leaks and identify the sources of the leaks, Republican strategist Michael McKenna said.

First, the Trump administration must gather information for every Cabinet secretary and top White House aide about any contact with Russian officials or anyone with ties to Moscow, with notice that failure to disclose is a firing offense, he said.

The president would then know the full extent of exposure on the issue and could decide what information to proactively release to the public. Otherwise, Mr. Trump can expect opponents to continue to release information.

“There is no hiding this stuff because they’ve obviously been listening to it for a while,” Mr. McKenna said. “They are intelligence guys. They have dossiers on all this stuff. They listen in on phone calls. They probably open mail. They are no doubt surveilling electronic traffic — that’s all stuff that folks in the White House don’t have.”

The allegations of a Russian connection have trailed Mr. Trump since the campaign, when U.S. intelligence agencies blamed Moscow for email hacks of the Democratic National Committee and the Hillary Clinton campaign — hacks that, they said, were intended to hurt Mrs. Clinton and therefore ultimately benefited Mr. Trump.

The House and Senate intelligence committees are investigating Russian meddling in the election. The committee’s Republican leaders have said they have yet to see evidence from intelligence agencies that the Trump campaign was in cahoots with Moscow.

Mr. Sessions recused himself Thursday from the FBI investigation into Moscow’s interference in the election amid a political firestorm over revelations that he met twice in his capacity as a U.S. senator with the Russian ambassador during the campaign.

Deepening the crisis for the White House, many Republican lawmakers joined Democrats in calling for Mr. Sessions to recuse himself, although they refrained from joining the opposition party’s calls for the attorney general to resign.

“Republican members of Congress have got to stop being afraid of their own shadow and falling on their sword at the first inference of bad news,” said Republican strategist Ford O’Connell. “They need to grow a backbone and see the bigger picture here and hold the line. They need to understand what Democrats want here. They want Trump out — impeached or resigned, they don’t care which.”

Mr. Trump also should demand his critics produce evidence to back up allegations of a Russian conspiracy within the administration, said Mr. McKenna.

“Start pushing back immediately,” he said. “The only way to deal with this is to stop shadow boxing it and saying, ‘Well, it’s not important. We should or shouldn’t recuse guys.’ Just go right after it.”

Mr. Trump began pushing back Saturday by tweeting about reports that the Obama administration sought wiretaps of Trump Tower during the campaign, saying President Obama resorted to “Nixon/Watergate” and “McCarthyism” tactics.

“Terrible! Just found out that Obama had my ‘wires tapped’ in Trump Tower just before the victory. Nothing found. This is McCarthyism!” Mr. Trump said in a series of tweets that have dominated headlines for days.

The president’s claims, apparently based on a variety of published reports about the Obama administration’s request for approval of wiretaps from the Foreign Intelligence Surveillance Act court, were widely dismissed in the news media for lacking proof.

Nevertheless, the wiretap requests for Trump Tower that were reportedly denied in June and approved in October underscore suspicions about the intelligence community.

“Just out: The same Russian Ambassador that met Jeff Sessions visited the Obama White House 22 times, and 4 times last year alone,” tweeted Mr. Trump, apparently referring to a Daily Caller report that chronicled Mr. Kislyak’s visits according to a White House log.

The intense criticism from Democrats spurred numerous reports of their meetings with Mr. Kislyak, including meetings that Sen. Claire McCaskill of Missouri and House Minority Leader Nancy Pelosi of California forgot about when they proclaimed that they had never met with him.

Still, the scrutiny of the Trump administration about Russia has begun to affect the president’s agenda. The president and his aides told diplomats and other officials that they are holding off on plans to make a deal with Russian President Vladimir Putin to jointly combat the Islamic State.

The president and his aides have ascribed the thinking to Moscow’s provocations, according to The Associated Press. But the backpedaling on a key tenet of his foreign policy highlights the political peril for Mr. Trump of appearing too close to Mr. Putin.

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